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The objective of this paper is to develop a semi-analytical finite element method for solving chemical dissolution-front instability problems in fluid-saturated porous media.

The porosity, horizontal and vertical components of the pore-fluid velocity and solute concentration are selected as four fundamental unknown variables for describing chemical dissolution-front instability problems in fluid-saturated porous media. To avoid the use of numerical integration, analytical solutions for the property matrices of a rectangular element are precisely derived in a purely mathematical manner. This means that the proposed finite element method is a kind of semi-analytical method. The column pivot element solver is used to solve the resulting finite element equations of the chemical dissolution-front instability problem.

The direct use of horizontal and vertical components of the pore-fluid velocity as fundamental unknown variables can improve the accuracy of the related numerical solution. The column pivot element solver is useful for solving the finite element equations of a chemical dissolution-front instability problem. The proposed semi-analytical finite element method can produce highly accurate numerical solutions for simulating chemical dissolution-front instability problems in fluid-saturated porous media.

Analytical solutions for the property matrices of a rectangular element are precisely derived for solving chemical dissolution-front instability problems in fluid-saturated porous media. The proposed semi-analytical finite element method provides a useful way for understanding the underlying dynamic mechanisms of the washing land method involved in the contaminated land remediation.

The purpose of this paper is to carry out a comprehensive study of compressible flow over double wedge and biconvex airfoils using computational fluid dynamics (CFD) and three analytical models including shock and expansion wave theory, Busemann's second‐order linearized approximation and characteristic method (CHM).

Flow over double‐wedge and biconvex airfoils was investigated by the CFD technique using the Spalart‐Allmaras turbulence model for computation of the Reynolds stresses. Flow was considered compressible, two dimensional and steady. The no slip condition was applied at walls and the Sutherland law was used to calculate molecular viscosity as a function of static temperature. First‐order upwind discretization scheme was used for the convection terms. Finite‐volume method was used for the entire solution domain meshed by quadratic computational cells. Busemann's theory, shock and expansion wave technique and CHM were the analytical methods used in this work.

Static pressure, static temperature and aerodynamic coefficients of the airfoils were calculated at various angles of attack. In addition, aerodynamic coefficients of the double‐wedge airfoil were obtained at various free stream Mach numbers and thickness ratios of the airfoil. Static pressure and aerodynamic coefficients obtained from the analytical and numerical methods were in excellent agreement with average error of 1.62 percent. Variation of the static pressure normal to the walls was negligible in the numerical simulation as well as the analytical solutions. Analytical static temperature far from the walls was consistent with the numerical values with average error of 3.40 percent. However, it was not comparable to the numerical temperature at the solid walls. Therefore, analytical solutions give accurate prediction of the static pressure and the aerodynamic coefficients, however, for the static temperature; they are only reliable far from the solid surfaces. Accuracy of the analytical aerodynamic coefficients is because of accurate prediction of the static pressure which is not considerably influenced by the boundary layer. Discrepancies between analytical and numerical temperatures near the walls are because of dependency of temperature on the boundary layer and viscous heating. Low‐speed flow near walls causes transformation of the kinetic energy of the free stream into enthalpy that leads to high temperature on the solid walls; which is neglected in the analytical solutions.

This paper is useful for researchers in the area of external compressible flows. This work is original.

Currently, there is a lack of fast and highly accurate on analytical solution of Reynolds equation for evaluating the characteristics of surface textured bearing. This paper aims to develop such an analytical solution of Reynolds equation for an effective analysis of the characteristics of surface textured bearings.

By using the separation of variables method and mean eigenvalue method, the analytical solution is constructed. The CFD simulations and experimental results are used to validate the correctness of the analytical solution.

The analytical solution can accurately evaluate the characteristics of textured bearings. It is found that the larger the eccentricity ratio and aspect ratio, the greater the oil film force. It also found that the smaller the eccentricity ratio, the larger the Sommerfeld number S. When eccentricity ratio e = 0.65, the attitude angles of different oil boundaries are same. The effect of different aspect ratios on dynamic stiffness and damping coefficient generally follows a same trend. It is numerically shown that the critical speed of rotor-bearing is 3500 rpm.

The analytical solution provides a simple yet effective way to study the characteristics of surface textured bearings.

The purpose of this paper is to introduce a closed-form analytical solution to evaluate the nominal moment capacity and associated deflections of steel-FRP beam systems. The proposed solution takes into consideration the partial composite behavior resulting from the interfacial contact and slip between the subcomponents of the system.

The partial composite action theory was used to develop an elastic analytical solution for the deflection of simply supported composite steel-FRP beams subjected to a mid-span point load. The solution takes into consideration the partial composite behavior of the system that arises from the interlayer slip at the steel-FRP interface.

The developed analytical model is used to predict the nominal moment capacity of the composite beam and the load value at the onset of yielding in the steel subcomponent of the section. The distribution of shear forces induced in the steel fasteners due to the interfacial slip is also obtained analytically. A comparative study is conducted by comparing the analytical results to their counterparts resulting from finite element modeling of the composite steel-FRP system. The agreement between analytical results and finite element predictions validates the accuracy of the derived analytical solution for partial composite steel-FRP beams.

The proposed solution applies only to the FRP strips and 6 mm steel bolts used in the study.

Recent studies revealed a promising efficiency of using mechanically fastened hybrid FRP sheets in strengthening steel beams. A major advantage of this technique is the ductile behavior of the steel-FRP system. The current paper introduces a closed-form analytical solution to evaluate the nominal moment capacity and associated deflections of steel-FRP beam systems. Forces developed at the steel-FRP interface due to the relative slip between both components are considered in the proposed analytical solution.

This paper presents approximate analytical solutions for the diffusion problems of a cylindrical hole in an infinite medium and a slot in an infinite medium with properly prescribed boundary conditions and initial conditions. These solutions have much simpler forms than those of exact analytical solutions, and asymptotically approach the exact solutions with increasing time or the material point moving away from the internal boundary. The approximate analytical solution for the diffusion problem of a slot in an infinite medium is applied to establish a shape function for the infinite elements. Good agreement is found in comparison of our results with those presented by Li and Huang and Cinco‐Ley et al. Finally, an example simulating a primary recovery procedure in hydraulic fracturing technique for an oil field is presented.

The first compressible flow solution based solely on the locally analytical method is developed. This is accomplished by developing the flow model and locally analytical solution for inviscid subsonic compressible flow. The stream function for irrotational, compressible flow without body forces was chosen as the governing differential equation. To demonstrate the modelling and locally analytical solution, this analysis is then applied to predict the flow in convergent nozzles, both planar and axially symmetric, for different back pressures. Results are presented which demonstrate the effectiveness of this technique.

The purpose of this study was to investigate the changes in solder joint stress when subjected to mechanical bending. The analytical theory pertaining to the stresses in the solder joint between the components (including the molding compound, the chip and the substrate) was described, and the printed circuit board (PCB) with a discontinuity function when the PCB assembly is subjected to mechanical bending was developed. Thus, the findings reported here may lead to a better understanding of the solder joint failure based on the Physics of Failure model.

This paper discusses the analytical model for calculating the stress in solder joints, as well as presents a simulation model that can be used for calculating the strain energy density of solder joint. First, the multilayer plate theory is used in discussing the composite material for the component, including the molding compound, the silicon chip and the substrate, or the PCB, including the copper layers, the fiber and the epoxy. Finally, the complete structure of the analytical model developed as a part of this current work is presented.

For the analytical model of multilayer structures in which the interconnection layer is discrete, mechanical bending has been modeled with respect to varying silicon chip length. The analytical model that describes the stress of the outermost solder joint experiences is chosen, as this is the typical solder joint failure. The analytical model can be applied to discrete solder joints, which are evaluated by calculating the matrix form. Owing to its use of the matrix equation, the analytical model can be highly combinatorial and thus more capable of calculating the solution.

The analytical solution based on a simple concept was presented and validated using the finite element model for the stress experienced by solder joints subjected to mechanical bending. To verify that the simulation represents a real PCB case, the authors use the finite element method (FEM) to compare their case with the multilayer plate theory. Owing to the good agreement between the theory and simulation results, the authors conclude that the multilayer plate theory can be correctly applied in multilayer PCB and be used in an analytical model for the PCB assembly subjected to mechanical bending.

Owing to the good agreement between the theory and simulation results, the authors conclude that the multilayer plate theory can be correctly applied in multilayer PCB and be used in an analytical model for the PCB assembly subjected to mechanical bending.

The analytical model is validated with the FEM model and provides the way to physically examine the solder joint failure mechanism. In this paper, the analytical model is developed as a means to assess the solder joint stress subjected to mechanical bending.

The analytical model treats the solder joint as discrete and has been successfully validated against the finite element model. The complete structure model (the second analytical model) is presented to discuss the effects of varying silicon chip length on the normal stress in solder joints. When the silicon chip length exceeds to 80 per cent of the total package length, the stress of the outermost solder joint increases rapidly. The design analysis findings have suggested that the failure of the outermost solder joint subjected to mechanical bending on the PCB assembly can be reduced by analyzing the analytical model.

The purpose of this paper is to develop a new transversal method of lines for one-dimensional reaction–diffusion equations that is conservative and provides piecewise–analytical solutions in space, analyze its truncation errors and linear stability, compare it with other finite-difference discretizations and assess the effects of the nonlinear diffusion coefficients, reaction rate terms and initial conditions on wave propagation and merging.

A conservative, transversal method of lines based on the discretization of time and piecewise analytical integration of the resulting two-point boundary-value problems subject to the continuity of the dependent variables and their fluxes at the control-volume boundaries, is presented. The method provides three-point finite difference expressions for the nodal values and continuous solutions in space, and its accuracy has been determined first analytically and then assessed in numerical experiments of reaction-diffusion problems, which exhibit interior and/or boundary layers.

The transversal method of lines presented here results in three-point finite difference equations for the nodal values, treats the diffusion terms implicitly and is unconditionally stable if the reaction terms are treated implicitly. The method is very accurate for problems with the interior and/or boundary layers. For a system of two nonlinearly-coupled, one-dimensional reaction–diffusion equations, the formation, propagation and merging of reactive fronts have been found to be strong function of the diffusion coefficients and reaction rates. For asymmetric ignition, it has been found that, after front merging, the temperature and concentration profiles are almost independent of the ignition conditions.

A new, conservative, transversal method of lines that treats the diffusion terms implicitly and provides piecewise exponential solutions in space without the need for interpolation is presented and applied to someone.

The purpose of this study is to investigate the post-buckling analysis of functionally graded columns by using three analytical, approximate and numerical methods. A pre-defined function as an initial assumption for the post-buckling path is introduced to solve the differential equation. The finite difference method is used to approximate the lateral deflection of the column based on the differential equation. Moreover, the finite element method is used to derive the tangent stiffness matrix of the column.

The non-linear buckling analysis of functionally graded materials is carried out by using three analytical, finite difference and finite element methods. The elastic deformation and Euler-Bernoulli beam theory are considered to establish the constitutive and kinematics relations, respectively. The governing differential equation of the post-buckling problem is derived through the energy method and the calculus variation.

An incremental iterative solution and the perturbation of the displacement vector at the critical buckling point are performed to determine the post-buckling path. The convergence of the finite element results and the effects of geometric and material characteristics on the post-buckling path are investigated.

The key point of the research is to compare three methods and to detect error sources by considering the derivation process of relations. This comparison shows that a non-incremental solution in the analytical and finite difference methods and an initial assumption in the analytical method lead to an error in results. However, the post-buckling path in the finite element method is traced by the updated tangent stiffness matrix in each load step without any initial limitation.

It is well known that the prefabricated vertical drain (PVD) installation process generates a significant soil disturbance around PVD. This disturbed zone significantly affects the rate of settlement and excess pore pressure dissipation. However, the characteristics of these zones were still uncertain and difficult to quantify; there remains large discrepancy among researchers. This study aims to develop a simple analytical solution for radial consolidation analysis of PVD-installed deposit considering mandrel-induced disturbance.

The proposed solution takes into account the nonlinear distributions of both horizontal hydraulic conductivity and compressibility toward the drain. The proposed solution was applied to analyze field behavior of test embankment in New South Wales, Australia.

Both effects significantly increased the time required to achieve a certain degree of consolidation. The effect of hydraulic conductivity on the consolidation rate was more significant than the effect of compressibility variation. And, the increased compressibility in the soil-disturbed zone due to mandrel installation significantly increased vertical strain of the PVD-improved soil deposit. The predicted results using the proposed analytical solution were in good agreement with the field measurements.

A geotechnical engineer could use the proposed analytical solution to predict consolidation behavior of drainage-installed ground.

Consolidation behavior of PVD-installed ground could be reasonably predicted by using the proposed solution with considering variations of both hydraulic conductivity and compressibility due to PVD installation.